Which composition of transformations will create a pair of similar, but not congruent, triangles?

The answer

The correct choice is the composition that contains a dilation, such as a rotation then a dilation (or reflection + dilation, translation + dilation, etc.). The key word is dilation: it is the only one of the four common transformations that changes a figure's size. Changing size is exactly what "similar but not congruent" requires.

Two triangles are similar when they have the same shape—equal corresponding angles and proportional corresponding sides. They are congruent when they are similar and the same size (a scale factor of 1). So to get "similar, not congruent," you need a transformation that preserves shape but alters size. That is precisely the definition of a dilation with a scale factor other than 1.

Rigid motions vs. dilations

Transformations fall into two families:

• Rigid motions (isometries): translations, rotations, and reflections. These preserve both size and shape. Distances and angles are unchanged, so the image is always congruent to the original.
• Non-rigid motions: dilations. A dilation multiplies every distance from a center point by a scale factor k. Angles are preserved (shape stays), but lengths scale by k, so unless k = 1 the size changes and the result is not congruent.

Because of this, any composition built only from translations, rotations, and reflections can never break congruence—no matter how many you stack. The moment you add a dilation with k ≠ 1, the triangles become similar but not congruent.

Why the other options are wrong

• Rotation then reflection — both are rigid motions, so the image is congruent. Same shape and same size. Not the answer.
• Translation then rotation — again, two rigid motions; congruent result. Not the answer.
• Reflection then translation — a glide-reflection-style combination, still entirely rigid; congruent. Not the answer.
• Rotation then dilation — the rotation reorients the triangle (still congruent so far), then the dilation rescales it. The final image has the same angles but different side lengths: similar, not congruent. This is the correct composition.

A quick test for any multiple-choice version of this problem: scan each option and ask, "Does it contain the word dilation?" Exactly one option will, and that is your answer.

The bigger picture

This distinction underlies how similarity is formally defined in geometry. Two figures are congruent if a sequence of rigid motions maps one onto the other, and similar if a sequence of rigid motions and dilations maps one onto the other. Congruence is therefore just a special case of similarity where the dilation factor equals 1. Understanding that a dilation is the only size-changer explains why it must appear whenever the question asks for similar-but-not-congruent, and why it must be absent whenever the question asks for congruent figures.